Resistance, resistivity, and Ohm’s law
Resistance
Describe the resistance of an object using physical properties of that object.
Resistance is a measure of the degree to which an object opposes the movement of electric charge.
The resistance of a resistor with uniform geometry is proportional to its resistivity and length and is inversely proportional to its cross-sectional area. \[\begin{equation} R = \dfrac{\rho l}{A} \end{equation}\]
Resistivity is a fundamental property of a material that depends on its atomic and molecular structure and quantifies how strongly the material opposes the motion of electric charge.
The resistivity of a conductor typically increases with temperature.
The total resistance of a resistor with uniform geometry, but that is made of a material whose resistivity varies along the length of the resistor, is given by \[\begin{equation} R = \int \dfrac{\rho(l) dl}{A} \end{equation}\]
Describe the electrical characteristics of elements of a circuit.
Ohm’s law relates current, resistance, and potential difference across a conductive element of a circuit. \[\begin{equation} I = \dfrac{\Delta V}{R} \end{equation}\]
Materials that obey Ohm’s law have constant resistance for all currents and are called ohmic materials.
The resistivity of an ohmic material is constant regardless of temperature.
Resistors can also convert electrical energy to thermal energy, which may change the temperature of both the resistor and the resistor’s environment.
The resistance of an ohmic circuit element can be determined from the slope of a graph of the current in the element as a function of the potential difference across the element
In addition to sources, a simple circuit might have passive components such as resistors, capacitors, and inductors.
Resistors
Resistors are devices that dissipate electrical energy. They are given the symbol \(R\) and are depicted on circuit diagrams as shown in figure \[fig:resistor\].
(0,2) to \[R=$R$,v=$e_R$, i=$i_R$\] (0,0);
The constitutive relationship for the resistor in figure \[fig:resistor\] is given by Ohm’s Law: \[\begin{equation} e_R = i_R R \quad \mbox{Ohm's Law} \end{equation}\] where \(R\) is a constant called the resistance. Resistance is measured in units of Ohms (Ω). Resistors are available in resistances from 1 Ω to several thousand (kΩ) or million (MΩ). The resistance value is marked as a series of colored bands (see figure \[fig:resistor-bands\]). Several colorful mnemonics (some dating to WWII and wildly inappropriate today) exist for learning the resistor color code. A meter may be used to check resistance values before installation, but the color bands are useful for quality assurance and verifying the correct resistors are installed in a circuit.
| black brown red orange yellow | 0 1 2 3 4 | green blue violet grey white | 5 6 7 8 9 |
| tolerances: no band silver gold | ± 20 % ± 10 % ± 5 % |
The power dissipated in a resistor (“Ohmic heating”) is given by the voltage times the current: \[\begin{equation} P = e i = \frac{e^2}{R} = i^2 R \end{equation}\]
A smal low-power resistor used as a discrete device on a circuit board might have a power rating of 1/8 or 1/4 W, while very large power resistors (often in tubular geometries with means for cooling) may be used for braking in motor drives and transport systems like electric-drive buses; or as heaters in plants or industrial processes.
Capacitors
Capacitors store electrical energy via an electric field between two charged plates with an insulating dielectric material between them. They are given the symbol \(C\) and are depicted on circuit diagrams as shown in figure \[fig:capacitor\].
(0,2) to \[C=$C$,v=$e_C$, i=$i_C$\] (0,0); (2,0) to \[polar capacitor=$C$, v\>=$e_C$, i\<=$i_C$\] (2,2);
The consitutive relationship for a capacitor is given by: \[\begin{align} Q &= C e_C \quad \text{or alternatively,} \\ i_C &= C \frac{de_C}{dt} \quad \text{or} \\ e_C &= \frac{1}{C} \int_0^t i_C dt \quad \text {in integral form} \end{align}\] where \(Q\) is charge and \(C\) is the capacitance, measured in Farads (\(\qty{1}{\farad} = \qty{1}{\coulomb}/\qty{1}{\volt}\)). The Farad is a large unit, typical capacitors are on the order of \(\qty{1e-12}{\farad} = \qty{1}{\pico\farad}\) to \(\qty{1e-6}{\farad} = \qty{1}{\micro\farad}\). For a parallel plate capacitor, the capacitance can be computed using \(C=\frac{\epsilon A}{d}\) where \(\epsilon\) is the dielectric constant, \(A\) plate area, and \(d\) plate spacing.
The energy stored in a capacitor is given by \(E=\frac{1}{2} C V^2\). The energy storage property of capacitors makes them useful in implementing analog filters, integrators and differentiators; passing AC while blocking DC; decoupling circuits from noise due to switching.
There are several types of capacitors. Ceramic disc capacitors look like disks (fig \[fig:capmarking\] top); while electrolytic capacitors often appear like cylindrical cans (fig \[fig:capmarking\] bottom). Some capacitors (notably, electrolytic) have polarity and can explode if installed backwards.
Ceramic disc capacitors are marked using a three-digit code expressed in pF. The first two digits correspond to the first two significant digits of the capacitance, while the third digit gives a power of ten multiplier. Thus “104” might signify \(\qty{10e4}{\pico\farad}=\qty{0.1}{\micro\farad}\).
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Inductors
Inductors store electrical energy via a magnetic field typically created by currents in a coil of wire. They are given the symbol \(L\) and are depicted on circuit diagrams as shown in figure \[fig:inductor\].
(0,2) to \[L=$L$,v=$e_L$, i=$i_L$\] (0,0);
The consitutive relationship for an inductor is given by: \[\begin{align} \lambda &= L i_L \quad \text{or alternatively,} \\ e_L &= L \frac{di_L}{dt} \\ \end{align}\] where \(\lambda\) is the flux linkage and \(L\) is the inductance, measured in Henrys (\(\qty{1}{\henry} = \qty{1}{\volt\second}/\qty{1}{\ampere}\)).
The energy stored in an inductor is given by \(E=\frac{1}{2}L I^2\). Inductors may be used to choke off high frequency noise at the inputs to systems, in some DC-DC converters, and in some engine ignition systems. Because of their construction of coils of wire around ferrous elements, motors, generators and transformers generally contain large inductances.
See also
- list here